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 A006530 Gpf(n): greatest prime dividing n, for n >= 2; a(1)=1. (Formerly M0428) 662

%I M0428

%S 1,2,3,2,5,3,7,2,3,5,11,3,13,7,5,2,17,3,19,5,7,11,23,3,5,13,3,7,29,5,

%T 31,2,11,17,7,3,37,19,13,5,41,7,43,11,5,23,47,3,7,5,17,13,53,3,11,7,

%U 19,29,59,5,61,31,7,2,13,11,67,17,23,7,71,3,73,37,5,19,11,13,79,5,3,41,83,7,17,43

%N Gpf(n): greatest prime dividing n, for n >= 2; a(1)=1.

%C The initial term a(1)=1 is purely conventional: The unit 1 is not a prime number, although it has been considered so in the past. 1 is the empty product of prime numbers, thus 1 has no largest prime factor. - _Daniel Forgues_, Jul 05 2011

%C Greatest noncomposite number dividing n. - _Omar E. Pol_, Aug 31 2013

%D M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 844.

%D D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section IV.1.

%D H. L. Montgomery, Ten Lectures on the Interface Between Analytic Number Theory and Harmonic Analysis, Amer. Math. Soc., 1996, p. 210.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Daniel Forgues, <a href="/A006530/b006530.txt">Table of n, a(n) for n=1..100000</a> [First 10000 terms from T. D. Noe]

%H M. Abramowitz and I. A. Stegun, eds., <a href="http://apps.nrbook.com/abramowitz_and_stegun/index.html">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

%H K. Alladi and P. Erdős, <a href="http://projecteuclid.org/euclid.pjm/1102811427">On an additive arithmetic function</a>, Pacific J. Math., Volume 71, Number 2 (1977), 275-294. MR 0447086 (56 #5401).

%H G. Back and M. Caragiu, <a href="http://www.fq.math.ca/Papers1/48-4/Back_Caragiu.pdf">The greatest prime factor and recurrent sequences</a>, Fib. Q., 48 (2010), 358-362.

%H A. E. Brouwer, <a href="/A046670/a046670.pdf">Two number theoretic sums</a>, Stichting Mathematisch Centrum. Zuivere Wiskunde, Report ZW 19/74 (1974): 3 pages. [Cached copy, included with the permission of the author]

%H Paul Erdős, Andrew Granville, Carl Pomerance and Claudia Spiro, <a href="http://math.dartmouth.edu/~carlp/iterate.pdf">On the normal behavior of the iterates of some arithmetic functions</a>, Analytic number theory, Birkhäuser Boston, 1990, pp. 165-204.

%H Paul Erdos, Andrew Granville, Carl Pomerance and Claudia Spiro, <a href="/A000010/a000010_1.pdf">On the normal behavior of the iterates of some arithmetic functions</a>, Analytic number theory, Birkhäuser Boston, 1990, pp. 165-204. [Annotated copy with A-numbers]

%H Nathan McNew, <a href="http://dx.doi.org/10.1080/10586458.2016.115518">The Most Frequent Values of the Largest Prime Divisor Function</a>, Exper. Math., 2017, Vol. 26, No. 2, 210-224.

%H OEIS Wiki, <a href="/wiki/Greatest_prime_factor_of_n">Greatest prime factor of n</a>

%H H. P. Robinson, <a href="/A006530/a006530.pdf">Letter to N. J. A. Sloane, Oct 1981</a>

%H David Singmaster, <a href="/A005178/a005178.pdf">Letter to N. J. A. Sloane</a>, Oct 3 1982.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GreatestPrimeFactor.html">Greatest Prime Factor</a>

%H <a href="/index/Cor#core">Index entries for "core" sequences</a>

%F a(n) = A027748(n, A001221(n)) = A027746(n, A001222(n)); a(n)^A071178(n) = A053585(n). - _Reinhard Zumkeller_, Aug 27 2011

%F a(n) = A000040(A061395(n)). - _M. F. Hasler_, Jan 16 2015

%F a(n) = n + 1 - Sum_{k=1..n}(floor((k!^n)/n) - floor(((k!^n)-1)/n)). - _Anthony Browne_, May 11 2016

%F n/a(n) = A052126(n). - _R. J. Mathar_, Oct 03 2016

%F If A020639(n) = n [when n is 1 or a prime] then a(n) = n, otherwise a(n) = a(A032742(n)). - _Antti Karttunen_, Mar 12 2017

%F a(n) has average order pi^2*n/(12 log n) [Brouwer]. See also A046670. - _N. J. A. Sloane_, Jun 26 2017

%p with(numtheory,divisors); A006530 := proc(n) local i,t1,t2,t3,t4,t5; t1 := divisors(n); t2 := convert(t1,list); t3 := sort(t2); t4 := nops(t3); t5 := 1; for i from 1 to t4 do if isprime(t3[t4+1-i]) then return t3[t4+1-i]; fi; od; 1; end;

%p # alternative

%p A006530 := n->max(1,op(numtheory[factorset](n))); # _Peter Luschny_, Nov 02 2010

%t Table[ FactorInteger[n][[ -1, 1]], {n, 100}] (* _Ray Chandler_, Nov 12 2005 and modified by _Robert G. Wilson v_, Jul 16 2014 *)

%o (PARI) A006530(n)=if(n>1,vecmax(factor(n)[,1]),1) \\ Edited to cover n=1. - _M. F. Hasler_, Jul 30 2015

%o (MAGMA) [ #f eq 0 select 1 else f[ #f][1] where f is Factorization(n): n in [1..86] ] // _Klaus Brockhaus_, Oct 23 2008

%o (Scheme)

%o ;; The following uses macro definec for the memoization (caching) of the results. A naive implementation of A020639 can be found under that entry. It could be also defined with definec to make it faster on the later calls. See http://oeis.org/wiki/Memoization#Scheme

%o (definec (A006530 n) (let ((spf (A020639 n))) (if (= spf n) spf (A006530 (/ n spf)))))

%o ;; _Antti Karttunen_, Mar 12 2017

%Y Cf. A020639 (smallest prime divisor), A034684, A028233, A034699, A053585. See also A032742, A052126, A070087, A070089, A061395, A175723.

%Y Cf. A046670 (partial sums), A104350 (partial products).

%Y See A124661 for "popular" primes.

%K nonn,nice,easy,core

%O 1,2

%A _N. J. A. Sloane_

%E Edited by _M. F. Hasler_, Jan 16 2015

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